Creating big time crystals with ultracold atoms

We first consider a single atom bouncing in the vertical direction z on a harmonically oscillating hard-wall mirror in the presence of strong transverse harmonic confinement. Introducing gravitational units ${l}_{0}={\left({\hslash }^{2}/\left({m}^{2}g\right)\right)}^{1/3}$, E₀ = mgl₀, ${t}_{0}=\hslash /\left(mg{l}_{0}\right)$ and assuming a one-dimensional (1D) approximation, the Hamiltonian of the system in the laboratory frame can be written

$$\begin{equation}H={p}^{2}/2+\tilde {V}\left[z+\left({\lambda}/{\omega }^{2}\right)z\enspace \mathrm{cos}\left(\omega t\right)\right]+z,\end{equation} \tag{ 1 }$$

where the potential (z ⩽ 0) = ∞ and (z > 0) = 0 for the hard-wall mirror, ω and λ/ω² are the frequency and the amplitude of the oscillating mirror in the laboratory frame, respectively, and m and g are the atom mass and gravitational acceleration. Use of gravitational units allows calculations to be performed with dimensionless parameters that are independent of the mass of the atom.

Transforming from the laboratory frame to the oscillating frame of the mirror, equation (1) becomes [5, 35]

$$\begin{equation}H={p}^{2}/2+\tilde {V}\left(z\right)+z+\lambda z\enspace \mathrm{cos}\left(\omega t\right),\end{equation} \tag{ 2 }$$

where λ is the amplitude of the time-periodic perturbation in the oscillating frame (hereafter referred to as the amplitude of the oscillating mirror). The case of a realistic soft Gaussian potential mirror is considered in section 2.6.

In the classical description, transforming to action-angle variables (I, θ)

$$\begin{equation}z=1/2{\left(3{\pi}I\right)}^{2/3}\left[1-{\left(\frac{\theta }{{\pi}}\right)}^{2}\right];\quad p=-{\left(\frac{3I}{{{\pi}}^{2}}\right)}^{1/3}\theta ,\end{equation} \tag{ 3 }$$

where 0 ⩽ θ < 2π, and using the secular approximation [35, 36], equation (2) can be expressed as a single-particle time-independent Hamiltonian [6]

$$\begin{equation}{H}_{\mathrm{F}}\approx {P}^{2}/\left(2{m}_{\mathrm{e}\mathrm{ff}}\right)+\lambda \langle z\rangle \mathrm{cos}\left(s{\Theta}\right),\end{equation} \tag{ 4 }$$

where Θ = θ − ωs/t, P = I − I_s, ${I}_{s}={\pi }^{2}{s}^{3}/\left(\right.3{\omega }^{3}$) is the action of the s : 1 resonant orbit. The effective mass and classical average value of z are given by

$$\begin{equation}{m}_{\mathrm{e}\mathrm{ff}}=-\frac{{\pi }^{2}{s}^{4}}{{\omega }^{4}}=-{\left(\frac{9}{\pi }\right)}^{2/3}{I}_{s}^{4/3};\quad \langle z\rangle =\frac{1}{{\omega }^{2}}=\frac{1}{{s}^{2}}{\left(\frac{3{I}_{s}}{{\pi }^{2}}\right)}^{2/3},\end{equation} \tag{ 5 }$$

where s = ω/Ω and Ω is the bounce frequency of the unperturbed particle (i.e., for a static mirror). The height of the classical turning point (drop height) of the bouncing atom is then h₀ = π²/(2Ω²).

Equation (4) is the effective Hamiltonian of a particle in the frame moving along an s : 1 resonant orbit and indicates that for s ≫ 1 a single resonantly driven atom in the vicinity of a resonant trajectory behaves like an electron moving in a crystalline structure created by ions in a solid state system. That is, eigenvalues of the quantized version of the Hamiltonian (4) form energy bands (see figure 1) and the corresponding eigenstates are Bloch waves. These eigenvalues are actually quasi-energies of a periodically driven particle while the eigenstates are Floquet states obtained in the frame moving along the resonant orbit [5, 6, 26, 35]. In the quantum description we will apply the tight-binding approximation by restricting the analysis to the first energy band of the quantized version of the Hamiltonian (4). In the many-body case such an approximation is valid provided the interaction energy per particle is smaller than the energy gap ΔE between the first and second energy bands shown in figure 1.

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We estimate the largest λ for a hard-wall potential mirror that is allowed before the dynamics become chaotic, making use of the Chirikov criterion [37]. This criterion estimates the value of λ for which two neighbouring resonance islands, s : 1 and (s + 1) : 1, described by equation (4), overlap. The distance between the islands (for s ≫ 1) and the half-width of the islands are

$$\begin{equation}{I}_{s+1}-{I}_{s}\approx \left({{\pi}}^{2}/{{\omega}}^{3}\right){s}^{2};\quad \frac{1}{2}{\left({\Delta}I\right)}_{s}=2\sqrt{\left\vert {m}_{\mathrm{e}\mathrm{ff}}{\lambda}\langle z\rangle \right\vert }=\sqrt{{\lambda}}2{\pi}{s}^{2}/{{\omega}}^{3}.\end{equation} \tag{ 6 }$$

The resonance islands nearly overlap when

$$\begin{equation}\frac{2}{3}\left({I}_{s+1}-{I}_{s}\right)\approx \frac{1}{2}{\left({\Delta}I\right)}_{s+1}+\frac{1}{2}{\left({\Delta}I\right)}_{s},\end{equation} \tag{ 7 }$$

which leads to λ ≈ (π/6)² ≈ 0.27. The 2/3 factor in (7) is an empirical correction that allows for the presence of higher-order resonances [35]. This means that for s ≫ 1 the critical value of λ is a constant independent of s and ω. For λ = 0.2, we find that, although there is some chaos between the neighbouring s : 1 and (s + 1):1 islands, the islands themselves are still not perturbed [6]. For λ < 0.2, the resonance islands become smaller and less suitable for realization of a time crystal. Indeed, quantum states that describe time crystals are located inside the resonance islands and if the islands are too small we need to choose a large value of I_s in order to realize a time crystal. While this is in principle possible, the resulting evolution of ultracold atoms that demonstrates time crystal behaviour becomes very long, see section 2.4. We conclude that for a hard-wall potential mirror λ = 0.2 is universally good for any s : 1 resonance for which s ≫ 1.

The quantum secular approximation allows us to obtain the quantum version of the classical Hamiltonian (4) [6]. That is, switching to the oscillating frame by means of the unitary transfomation $U={\mathrm{e}}^{\mathrm{i}\hat{n}{\omega}t/s}$, the matrix elements of the time-averaged quantum Hamiltonian describing the s : 1 resonance dynamics reads

$$\begin{equation}\langle {n}^{\prime }\left\vert {H}_{\mathrm{F}}\right\vert n\rangle \approx \left({E}_{n}-\frac{n{\omega}}{s}\right){\delta }_{{n}^{\prime },n}+\frac{\lambda }{2}\langle {n}^{\prime }\left\vert z\right\vert n\rangle \left({\delta }_{{n}^{\prime },n+s}+{\delta }_{{n}^{\prime },n-s}\right),\end{equation} \tag{ 8 }$$

where the |n⟩ are eigenstates of the unperturbed part of the Hamiltonian (2), i.e., H with λ = 0, with the corresponding eigenvalues E_n, and $\hat{n}\vert n\rangle =n\vert n\rangle$. The validity of the quantum secular approximation (8) can be checked for a given set of parameters by comparing results of the classical secular approach with the exact classical approach. That is, if the classical secular and exact treatments agree, the quantum secular approach is also valid [6]. Around the resonant value of the quantum number of the unperturbed particle, i.e., for n ≈ n₀ ≈ I_s, the first term on the right hand side of (8) can be approximated by

$$\begin{equation}{E}_{n}-\frac{n{\omega}}{s}\approx {E}_{{n}_{0}}-\frac{{n}_{0}{\omega}}{s}+\frac{{\left(n-{n}_{0}\right)}^{2}}{2{m}_{\mathrm{e}\mathrm{ff}}}.\end{equation} \tag{ 9 }$$

We know how the parameters of equation (4) scale with I_s. We now investigate how the effective mass m_eff, obtained from the quantum approach (8), and the matrix element ⟨n₀ − s/2|z|n₀ + s/2⟩, which provides an estimate of the classical average value ⟨z⟩, scale with n₀ and s. The results, presented in figure 2, show that for s ⩽ 100, the classical scaling is reproduced in the quantum approach if the particle quantum number n₀ ≳ 100. This allows us to use the classical analytical expressions to determine the optimal parameters for an experiment.

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To demonstrate that a time crystal is created one needs to show that the ultracold atoms evolve periodically with a period s-times longer than the driving period T = 2π/ω. If the interactions between atoms are too weak, the subharmonic periodic evolution will be destroyed because the atoms will start to tunnel between lattice sites of the potential in (4), or in other words between wave-packets propagating along the resonant orbit. Therefore, to demonstrate that a time crystal is created, the system needs to have evolved for at least the time period corresponding to the tunnelling time t_tunnel of a single atom between neighouring lattice sites. The tunnelling time (for s ≫ 1) and the bounce period T_bounce of the atom are given by [6]

$$\begin{equation}{t}_{\mathrm{t}\mathrm{u}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{l}}\approx 2.4/J;\quad {T}_{\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{e}}=2{\pi}/{\Omega}\end{equation} \tag{ 10 }$$

where J is the tunnelling amplitude of the particle between neighbouring sites of the periodic potential in (4), ${\Omega}={\left[{{\pi}}^{2}/\left(3{n}_{0}\right)\right]}^{1/3}$, and I_s is denoted here by the quantum number of the unperturbed particle, i.e., I_s ≈ n₀. We require t_tunnel to be as short as possible because then the number of bounces needed to demonstrate that a time crystal is robust against single-particle tunnelling is relatively small—each bounce off the mirror is a potential source of loss of atoms from the BEC. The number of bounces required to observe quantum tunnelling for non-interacting particles is then

$$\begin{equation}{N}_{\mathrm{b}}=\frac{{t}_{\mathrm{t}\mathrm{u}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{l}}}{{T}_{\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{e}}}\approx 2.4{\Omega}/\left(2{\pi}J\right)\quad for\enspace s\gg 1.\end{equation} \tag{ 11 }$$

Another important parameter that we wish to control is the energy gap ΔE between the first and second bands of the quantum version of the Hamiltonian (4), see figure 1. To realize a time crystal, sufficiently strong interactions between atoms need to be be present. However, if the interactions are too strong, the single-band description within the Bose–Hubbard model (see section 2.5) is no longer valid because higher bands of the Hamiltonian (4) become involved. Thus, ΔE should be as large as possible but this requirement is in contradiction with the requirement of a short tunnelling time t_tunnel and we need to find a compromise.

Dividing the classical Hamiltonian (4) by λ⟨z⟩ and rescaling sΘ → Θ, gives

$$\begin{equation}{H}_{\mathrm{F}}\approx {P}^{2}/\left[2{m}_{\mathrm{e}\mathrm{ff}}{\lambda}\langle z\rangle /{s}^{2}\right]+\mathrm{cos}\enspace {\Theta}={P}^{2}/\alpha +\mathrm{cos}\enspace {\Theta},\end{equation} \tag{ 12 }$$

which depends only on the single universal parameter

$$\begin{equation}{\alpha}=2\vert {m}_{\mathrm{e}\mathrm{ff}}\vert {\lambda}\langle z\rangle /{s}^{2}=2{{\pi}}^{2}{s}^{2}{\lambda}/{\omega }^{6}.\end{equation} \tag{ 13 }$$

The division by λ⟨z⟩ also means that in order to analyse the scaling properties of the system the quasi-energy gap ΔE and the tunnelling amplitude J should be expressed in units of = λ⟨z⟩ = λ/ω².

Figure 3 (top panel) presents calculations of the energy gap ΔE/J, obtained from the quantum secular Hamiltonian (8), cf figure 1, versus the universal parameter α for a range of values of s and n₀ for a hard-wall potential mirror. We see that for different values of s and n₀ all the quantum data lie on a single universal curve. Similarly, in figure 3 (bottom panel), the tunnelling amplitude J, in units of = λ⟨z⟩, obtained from diagonalization of the effective Hamiltonian (4) and from the fully quantum secular approach (8), versus α for different values of s and n₀ all lie on the same universal curve which we denote by $f\left({\alpha}\right)=J\left({\alpha}\right)/\varepsilon$. These results demonstrate that the scaling deduced from our analysis is valid and can be used for further purposes.

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With the help of f(α) = J(α)/, we can now express the number of bounces as

$$\begin{equation}{N}_{\mathrm{b}}=\frac{{t}_{\mathrm{t}\mathrm{u}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{l}}}{{T}_{\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{e}}}=2.4/\left[\sqrt{2{\lambda}{\alpha}}f\left({\alpha}\right)\right],\end{equation} \tag{ 14 }$$

which indicates that if we choose λ (e.g., the optimal value λ = 0.2 for a hard-wall potential mirror) and a value for the energy gap ΔE(α)/J(α) (which determines α) the number of bounces N_b is constant and independent of the values of s that we choose. In other words, for fixed λ and α we can choose any s : 1 resonance (then from α = constant we obtain n₀ and consequently ω) and the number of bounces is always the same. This is illustrated in the left panel of figure 4 where for different values of λ we always choose α so that the gap ΔE/J between the first and second energy bands of the quantum version of the Hamiltonian (4) is about 10 (which for λ = 0.2 corresponds to α = 0.456).

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The key results of the analysis are presented in the right panel of figure 4 where the number of bounces N_b needed for tunnelling of non-interacting atoms between lattice sites of the potential in (4) versus the band gap ΔE/J are presented. The smaller the band gap we can afford, the smaller the optimum number of bounces.

We now switch to the case of a BEC of interacting bosonic atoms bouncing on an oscillating mirror in the presence of strong transverse harmonic confinement ω_⊥ and assume the 1D approximation. Note that we focus on a spatially finite 1D system where for sufficiently weak interactions, the BEC is not destroyed by long-wavelength quantum fluctuations. We restrict the analysis to the Hilbert subspace corresponding to the first energy band of the quantum version of the Hamiltonian (4). This is the resonant subspace where atoms occupy s localized wavepackets w_i(z, t) evolving along the s : 1 resonant orbit with period sT, where T is the driving period [6]. These wave-packets are the time-periodic version of the Wannier states in solid state physics. Within the mean-field approach and restricting to the s-dimensional resonant Hilbert subspace we can expand solutions of the Gross–Pitaevskii equation in terms of localized Wannier-like states $\psi \left(z,t\right)=\sum _{i=1}^{s}{a}_{i}{w}_{i}\left(z,t\right)$ and obtain the energy functional (actually the quasi-energy functional) in the form [6]

$$\begin{equation}E\approx -\frac{1}{2}\sum _{i,j=1}^{s}{J}_{ij}\left({a}_{i}^{{\ast}}{a}_{j}+\mathrm{c}.\mathrm{c}.\right)+\frac{1}{2}\sum _{i,j=1}^{s}{U}_{ij}{\left\vert {a}_{i}\right\vert }^{2}{\left\vert {a}_{j}\right\vert }^{2},\end{equation} \tag{ 15 }$$

with

$$\begin{equation*}{J}_{{j}^{\prime }j}=-\frac{2}{sT}{\int }_{0}^{sT}\enspace \mathrm{d}t{\int }_{0}^{\infty }\mathrm{d}z{w}_{{j}^{\prime }}^{{\ast}}\left[\frac{{p}^{2}}{2}+z+\lambda z\enspace \mathrm{cos}\enspace \omega t-\mathrm{i}{\partial }_{t}\right]{w}_{j},\end{equation*}$$

$$\begin{equation*}{U}_{ii}=\frac{{g}_{\mathrm{1}\mathrm{D}}N}{sT}{\int }_{0}^{sT}\enspace \mathrm{d}t{\int }_{0}^{\infty }\enspace \mathrm{d}z{\left\vert {w}_{i}\right\vert }^{4},\end{equation*}$$

$$\begin{equation*}{U}_{ij}=\frac{2{g}_{\mathrm{1}\mathrm{D}}N}{sT}{\int }_{0}^{sT}\enspace \mathrm{d}t{\int }_{0}^{\infty }\enspace \mathrm{d}z{\left\vert {w}_{i}\right\vert }^{2}{\left\vert {w}_{j}\right\vert }^{2},\quad for\enspace i\ne j,\end{equation*}$$

where N is the total number of atoms, g_1D = 2ω_⊥a_s describes the contact interaction between the atoms, a_s is the s-wave scattering length (in gravitational units) and the U_ii and U_ij describe the on-site and long-range interaction energies per particle in the Bose–Hubbard model (15). The leading tunnelling amplitudes correspond to the nearest-neighbour hopping J = |J_i,i+1| and only this hopping can be kept in the model when we want to describe time crystal dynamics. Such a tight-binding approximation is valid provided the interaction energy per particle is much smaller than the energy gap ΔE between the first and second bands of the effective Hamiltonian (4), see figure 1.

In the left panel of figure 5 the lowest energy solutions of the mean-field energy (15) for s = 30, λ = 0.2, ω = 4.45 and different values of the interaction strength g_1DN are presented. We emphasize that these solutions are related to the lowest quasi-energy of the driven system within the resonant Hilbert subspace; they do not correspond to the ground state which does not exist in a periodically driven system. For g_1DN > −0.0024, the lowest energy solution is a uniform superposition of all s Wannier-like wave-packets, ${\psi}=\sum _{i=1}^{s}{w}_{i}/\sqrt{s}$. The wavefunction ψ evolves with the driving period T and no spontaneous breaking of discrete time-translation symmetry of the many-body Hamiltonian takes place. However, if the attractive interactions are sufficiently strong, it becomes energetically favourable for the atoms to localize, and the lowest energy mean-field solution reveals spontanoues breaking of time-translation symmetry; i.e., ψ evolves with a period s-times longer than T. The stronger the interactions, the better the localization of the atoms, and for g_1DN ⩽ −0.07, we obtain ψ ≈ w_i with the squared-overlap |⟨ψ|w_i⟩|² > 0.9. The corresponding interaction energy per particle |U_ii|/(2J) ≈ 0.8 (where J = |J_i,i+1|) is smaller than the energy gap ΔE/J = 10 between the first and second bands of the quantum version of the Hamiltonian (4), and consequently the Bose–Hubbard model (15) is valid. In the right panel of figure 5, the order parameter

$$\begin{equation}\mathit{O}=\frac{\mathrm{M}\mathrm{a}{\mathrm{x}}_{\mathrm{i}}\left(\vert \langle {\psi}\vert {\mathit{w}}_{\mathit{i}}\rangle {\vert }^{2}\right)-1/\mathit{s}}{\mathrm{1}-\mathrm{1}/\mathit{s}},\end{equation} \tag{ 16 }$$

which characterizes the overlap of the lowest energy solution ψ with a single Wannier-like wave-packet w_i versus the interaction strength g_1DN is presented for different s. The critical values of the interaction strength can be identified in the plots. The figure also shows how strong the interactions need to be in order to be dealing with the lowest energy solution that practically reduces to a single Wannier-like wave-packet w_i. This is important information because time crystals in which ψ(z, t) ≈ w_i(z, t) can be easily prepared in an experiment, see section 4. The case s = 40 has been extensively analysed in [6], where the numerical integration of the full Gross–Pitaevskii equation confirmed the results based on the Bose–Hubbard model.

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For different s:1 resonances, the critical interaction strength g_1DN differs because the tunnelling amplitudes J_ij are slightly different and also because the same g_1DN does not necessarily mean the same interaction coefficients U_ij in (15). The coefficients U_ij depend on the longitudinal width of the atom cloud which, for the optimal values of the parameters, varies from σ_z = 1.3–2.7 for s = 10–100 (table 2). Thus, a slightly different interaction parameter g_1DN and transverse confinement frequency ω_⊥ is required for different s.

The calculations in previous sections were based on a simple hard-wall mirror potential. We now consider the case of a realistic soft Gaussian mirror potential, such as that produced by a repulsive light-sheet,

$$\begin{equation}V\left(z\right)={V}_{0}\mathrm{exp}\left(-{z}^{2}/2{\sigma }_{0}^{2}\right),\end{equation} \tag{ 17 }$$

where V₀ and σ₀ are the height and width of the Gaussian mirror potential.

For a hard-wall potential mirror, the trajectories of the bouncing atoms reverse their direction abruptly at the reflection point (i.e., p → −p), so that the Fourier transform of the unperturbed periodic trajectories $z\left(t\right)=\sum _{k}{z}_{k}{\mathrm{e}}^{\mathrm{i}k{\Omega}t}$ (where Ω is the frequency of the bouncing atom in the absence of mirror oscillations, see (10)) results in amplitudes of the harmonics that decrease with k like z_k ∼ 1/k². In the case of a soft Gaussian potential mirror, the trajectories of the atoms are smoothly reflected, so that the harmonics decrease much faster with k, see left panel of figure 6. Because for an s : 1 resonance the amplitude of the potential in (4) is determined by the amplitude of the sth harmonic, i.e., ⟨z⟩ ≈ z_s, the Gaussian potential mirror needs to oscillate with an amplitude λ much larger than the oscillation amplitude of a hard-wall potential mirror in order to have the same effect on the bouncing atoms.

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When a particle bounces off a soft Gaussian potential mirror some harmonics of the classical unperturbed orbits are not created at all. The question of which harmonics disappear for a given V = V₀/E_particle ⩾ 1 (where E_particle is the particle's energy) is a complex problem. In the left panel of figure 6 we see dramatic drops of certain Fourier components which have a uniform spacing increasing from Δk = 16–33 for V ≈ 1.6–5. We focus here on the 30 :1 resonance. The right panel of figure 6 shows the amplitude z₃₀ of the 30th harmonic of an unperturbed periodic orbit as a function of V = V₀/E_particle. When V approaches one, i.e., V₀ = E_particle, from above there is a series of zeros of z₃₀. However, for V ≳ 10, there is no dramatic drop of |z₃₀| and an experiment demonstrating a discrete time crystal can be carried out in this regime.

For a given choice of V₀ and σ₀, our primary restriction is to obtain an energy gap of ΔE/J ≈ 10. First, we choose a mirror oscillation frequency ω and by applying the quantum secular approximation (8) we determine the optimal oscillation amplitude λ that leads to ΔE/J ≈ 10. We then need to check if the secular approximation is still valid for these parameters by examining the classical phase-space pictures of the action I versus angle Θ to see if the dynamics is still regular or if it is already chaotic.

We assume here a soft Gaussian mirror potential which corresponds approximately to the repulsive light-sheet mirror used in [38] for the reflection of a⁸⁷Rb BEC dropped from heights up to 300 μm. The mirror is formed by a ⩽ 3W, 532 nm laser beam with a waist σ₀l₀ = 10 μm and horizontal extension 200 μm, which for ³⁹K atoms corresponds to σ₀ = 15.5 and V_max ≈ 4.6 × 10³ in gravitational units. The hardness of the mirror can be varied by varying the beam waist σ₀. For a given Gaussian width σ₀ = 15.5, different mirror potential heights V₀/V_max, and oscillation frequencies around the optimal hard-wall mirror value ω = 4.45, we have used the quantum secular approach (8) to determine the amplitude of the mirror oscillations λ required to obtain ΔE/J ≈ 10. The results are shown in the left panel of figure 7. Now that we have predictions for all parameters, the corresponding classical phase-space pictures have been obtained, examples of which are shown in figure 8 for different mirror heights V₀/V_max. For ω = 4.45, the phase space around the 30 : 1 resonance islands is regular in all cases except V₀/V_max ≈ 0.2, for which the λ needed to obtain ΔE/J ≈ 10 is extremely high (>100) and the classical motion is no longer regular, and consequently the 30:1 resonance for V₀/V_max ≈ 0.2 is not suitable for realization of a time crystal. This is because for V₀/V_max ≈ 0.2 (and for a particle energy that fulfills V = V₀/E_particle ≈ 5), the 30th harmonic of the resonant periodic orbit disappears (figure 6). For V₀/V_max > 0.2, the drop height h₀ does not change too much as a function of V₀/V_max (figure 7, right panel) and the optimal number of bounces required for tunnelling to neighbouring wave-packets and the optimal tunnelling amplitude remain nearly constant at N_b ≈ 54 and J ≈ 0.0010 for ω = 4.45. All of these parameters are very close to the optimal values for the hard-wall mirror.

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For ω > 4.45 and λ corresponding to ΔE/J = 10, the classical motion becomes more chaotic than for ω = 4.45 and the secular approach is not fully valid, while for ω < 4.45 the number of bounces N_b required for tunnelling to neighbouring wave-packets increases. Thus, the optimal value of ω for the soft Gaussian potential mirror is close to the optimal value ω = 4.45 for the hard-wall potential mirror for s = 30.

The relatively large optimal values of λ in the case of the soft Gaussian potential mirror, e.g., λ = 2.3 (corresponding to λl₀/ω² = 74 nm for ³⁹K in the laboratory frame) at V₀/V_max = 0.8, ω = 4.45, required to obtain a band gap ΔE/J = 10 are more readily accessible experimentally than the optimal hard-wall mirror value λ = 0.2 (6.5 nm for ³⁹K). If still larger mirror oscillation amplitudes λ are required in the laboratory, one could operate closer to the peak of the lambda versus mirror potential height curve in figure 7, left panel, provided the corresponding classical resonance islands are not destroyed by chaos.

To operate in the quasi-1D regime, the transverse standard deviation σ_⊥ of the atomic density needs to be less than or equal to the standard deviation σ_z along the longitudinal direction and the interaction energy per particle should not exceed the excitation energy in the transverse directions (σ_⊥ and σ_z are defined as the Gaussian fits to the density of the atomic cloud). The transverse width is determined by the frequency ω_⊥ of the harmonic potential in the transverse directions, i.e., ${\sigma }_{\perp }=\sqrt{\hslash /\left(2m{{\omega}}_{\perp }\right)}$. These requirements imply that we need σ_z ⩾ σ_⊥ and ${\sigma }_{z}{ >}\left\vert {a}_{s}\right\vert N$, where a_s is the atomic s-wave scattering length. When these criteria are fulfilled, a BEC of ultracold atoms is also stable against 'bosenova' collapse that can occur for attractive interactions in three-dimensional space. For atoms bouncing resonantly on an oscillating mirror, the longitudinal width is smallest at the classical turning point and corresponds to σ_z = 1.81 for s = 30. For ³⁹K atoms, s = 30, ω = 4.45 and a Gaussian mirror with V₀/V_max = 0.8, σ₀l₀ = 10 μm, the minimum longitudinal width is σ_zl₀ = 1.17 μm; for example, for a_s l₀ = −1.7a₀, the maximum number of atoms is N_max ≈ 10 000.

To minimize losses due to three-body recombination, the mean-square atom density ⟨n²⟩ at the classical turning point needs to be less than 1/(K₃τ_BEC), where K₃ is the three-body recombination constant and τ_BEC is the lifetime of the BEC. For ³⁹K |1,+1⟩ atoms and taking K₃ = 1.3 (5) × 10⁻²⁹ cm⁶ s⁻¹ near the zero-crossing point [39] and τ_BEC ≈ 1 s, we obtain ⟨n²⟩_max ≈ 10²⁹ cm⁻⁶; for example, for σ_zl₀ = σ_⊥l₀ = 1.17 μm at the classical turning point, the maximum number of atoms is N_max ≈ 16 000.

The lifetime of a bouncing BEC may be limited by losses due to three-body recombination collisions, two-body collisions with background atoms and molecules, photon scattering caused by stray near-resonant light, and atoms missing the mirror due to spreading of the wave-packet in the transverse directions or walk-off.

The lifetime due to three-body recombination is given by τ_3b = [⟨n²⟩K₃]⁻¹, where K₃ = 1.3 (5) × 10⁻²⁹ cm⁶ s⁻¹ for ³⁹K |1, +1⟩ atoms near the zero-crossing point [40]; for example, for σ_zl₀ = σ_⊥l₀ = 1.17 μm (at the classical turning point) and N = 5000 atoms, we obtain ⟨n²⟩ ≈ 8 × 10²⁷ cm⁻⁶ and hence τ_3b ≈ 10 (4) s. Losses due to collisions with background atoms can be kept to a negligible level by maintaining a high-quality vacuum (<10⁻¹¹ millibar) and losses due to photon scattering should be negligible for light from the far-detuned 532 nm light-sheet mirror. Spreading of the atom cloud along the transverse directions will be suppressed due to the presence of the vertical optical waveguide. For atoms bouncing resonantly on an oscillating atom mirror, there will be no spreading in the longitudinal direction because each bounce from the oscillating mirror refocuses the atoms back [35]. In the calculations presented here, we assume about 50 bounces (corresponding to 1500 mirror oscillations for s = 30 and lasting about 0.6 s) can be achieved without significant atom losses for drop heights of around 200 μm.

For s = 30, ω = 4.45 and ω_z/(2πt₀) = ω_⊥/(2πt₀) ≈ 95 Hz, the drop height hl₀ ≈ 145 μm is much larger than the longitudinal extension of the atom cloud (2σ_zl₀ ≈ 2–10 μm between the turning point and the atom mirror) and thus it should be possible to probe the position of the atom cloud with reasonable spatial resolution. Higher resolution could be achieved, if required, by choosing a higher s resonance, for which the drop height (in SI units) scales as s^4/3 (table 2).

For a soft Gaussian potential mirror with σ₀l₀ = 10 μm, V₀/V_max = 0.8, ω = 4.45, the optimal oscillation amplitude is λ = 2.3, or λl₀/ω² = 74 nm at a mirror oscillation frequency of 2.8 kHz for ³⁹K in the laboratory frame. Still larger mirror oscillation amplitudes could be used, if required, by operating closer to the peak in figure 7(a) provided the corresponding classical resonance islands remain stable. Our estimates show that perturbations due to mechanical vibrations transmitted by a typical optical table are negligible when we want to control the motion of the atom mirror to a few nanometers at oscillation frequencies of a few kilohertz.

In the present experiment, the s-wave scattering length a_s needs to be adjusted precisely to small negative values (e.g., a_sl₀ = −1.6a₀ for g_1DN = −0.2 with N = 5000) for the time crystal phase or to zero for the non-time crystal phase. For the broad ³⁹K |1, +1⟩ Feshbach resonance, tuning the scattering length to a precision of ±0.1a₀ requires the magnetic field to be stable to ±0.18 G which is much larger than the stray AC and DC magnetic fields achievable in the laboratory.